The powers of i is always equal to either one of these 4 numbers: 1, i , -1,-i. Here we will see why.

i⁰ = 1, this is because for any non-zero number, any number with exponent 0 is equal to 1.

i¹ = i, this is because any number with exponent 1 is equal to itself, in this case, i itself.

i² = -1, this can be explained by the definition of imaginary number, which says i is defined as 

i³ = -i, since i²=-1, i³ = i² * i = -1 * i = -i

i⁴ = 1 becausei⁴ =  i³ * i = -i * i = – *   = 1

We see that i⁴ = i⁰, so this is a cycle, so by prediction i⁵ equals to i, i⁶ equals -1, and so on…. and in fact that is true.

So i with the exponent n (any whole number from 0 to above) can be determined by the following rules:

Divide the exponent by 4 and find the remainder,

if the remainder is 0 (no remainder), then iⁿ equals 1;

if the remainder is 1, then iⁿ equals i;

if the remainder is 2, then iⁿ equals -1;

if the remainder is 3, then iⁿ equals -i.